Why does a ray of light, incident through centre of curvature of a spherical
mirror, retrace its path?
 

Solution
the ray of light incident through centre of curvature retrace its path, because the angle between normal and incident ray is 0 degree.
Thus using snell's law, we can conclude that angle of refraction will be 0 degree.
Thus as a result the reflected ray bounce back to the same medium with 0 degree angle of reflection and refracted ray retraces the path in opposite direction. 

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